By Manjul Bhargava (auth.), Claus Fieker, David R. Kohel (eds.)

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Additional resources for Algorithmic Number Theory: 5th International Symposium, ANTS-V Sydney, Australia, July 7–12, 2002 Proceedings

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49–105. Ser97. , Friedr. Vieweg & Sohn, Braunschweig, 1997, Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a foreword by Brown and Serre. Shl89. Alexandra Shlapentokh, Extension of Hilbert’s tenth problem to some algebraic number fields, Comm. Pure Appl. Math. 42 (1989), no. 7, 939–962. Shl92. Alexandra Shlapentokh, Hilbert’s tenth problem for rings of algebraic functions in one variable over fields of constants of positive characteristic, Trans. Amer.

For the sake of simplicity, assume that q is prime. In that case, λq−3 ≡ λ−2 (mod q). Moreover, thanks to the relation i j i+j+1 tˆ(φ(g λ ), φ(g λ )) = g λ , q−3 −2 it is easy by using addition chains to compute Λ = g λ = g λ . Remarking λab ab ˆ that t(φ(g ), φ(Λ)) = g we can now solve the CDH problem in G3 = G1 (and also in G2 ) with two applications of the pairing tˆ. As a consequence of this digression, we can now remark that the hardness of the Tate Diffie–Hellman problem implies that the Tate pairing is hard to invert when one side of the pairing is fixed.

193 (2000), no. 2, 463–500. Shl00b. Alexandra Shlapentokh, Hilbert’s tenth problem over number fields, a survey, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Amer. Math. , Providence, RI, 2000, pp. 107–137. Sil92. Joseph H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original. Vid94. Carlos R. Videla, Hilbert’s tenth problem for rational function fields in characteristic 2, Proc. Amer. Math.

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